Enumerating Wild Card Hands

by Cactus Kev

The distribution of all possible 5-card poker hands is well known.
With a standard deck of 52 cards, there are C(52,5), or 2,598,960 ways to
select five distinct cards (where order does not matter). If we
determine the hand value for each of those approximately 2.6 million poker hands,
we get the following table:

Five Card Poker Hands

40

Straight Flush

624

Four of a Kind

3,744

Full House

5,108

Flush

10,200

Straight

54,912

Three of a Kind

123,552

Two Pair

1,098,240

One Pair

1,302,540

High Card

2,598,960

TOTAL

It is this table that initially gave us our poker hand rankings.
Notice that the fewer the ways available to make a particular poker
hand, the higher its value. That's why a Straight Flush beats
a Four-of-a-Kind. There are only 40 ways to make a Straight Flush,
but 624 ways to make a Four-of-a-Kind. I have sometimes seen the above
table given where a Royal Flush is listed separately, and ranked
directly above a Straight Flush. In my opinion, this is unnecessary,
since a Royal Flush is simply a special case of a Straight Flush (it
just happens to be the highest Straight Flush possible). We don't
single out Four Aces with a King kicker as being the highest possible
Four-of-a-Kind hand, so there is no need to single out the Royal
Flush hand either. But some tables list it that way, so just be
aware that I've have combined the Royals with the other Straight Flushes.

Now, recently I got to thinking about wild card games, and I wondered
what the frequency table looked like when you counted Deuces as being
wild. Since the rules for wild card games are numerous and definately
not standardized, I used the following stipulations:

Five-of-a-Kind hands are allowed.

Wild cards can represent any card (no restrictions).

If a hand contains one or more wild cards, that hand's value
is the highest possible it can be.

Double-Ace and Triple-Ace Flushes are not allowed.

Rule 1 is included because most players like to have Five-of-a-Kind
hands when playing with wild cards. Rule 2 means that a wild card
can be anything -- it is not restricted in its value like the Joker
is in some games (i.e. can only be used as an Ace, or to complete
a Flush or Straight). Rule 3 is required because hands
with wild cards can have multiple values. For example, if you hold
two Kings, two wild Deuces, and a Jack, you can claim you have the
following five hands:

Four of a Kind (KKKKJ)

Full House (KKKJJ or JJJKK)

Three of a Kind (KKKJx)

Two Pair (KKJJx, AAKKJ, etc.)

One Pair (KKJxx)

Since the highest hand wins in most poker games, it makes sense to count
wild card hands as their highest possible value. So in the above example,
that hand would be counted as a Four-of-a-Kind. Rule 4 is included so
that a hand like A983 (all Spades) with a wild Deuce will be counted as
an AK983 Flush, and not an AA983 Flush (a double Ace-High Flush, which
is allowed in some home games).

My first thought was to search the internet and check poker books to
see if such a table already existed. The internet turned up nothing,
but the book "Scarne's Guide to Modern Poker" had the table I was looking for.
However, I never take such tables as gospel, and I wanted to compute
and verify the values myself (which is a good thing, as I soon learned
that Scarne's table was incorrect!) I also posted this question in
the rec.gambling.poker newsgroup, and a number of RPG'ers posted
results they had either found or computed. Even then, some of the
figures given were incorrect or inconsistant; so I decided to determine
the hand frequencies myself.

There were a couple of ways I could tackle this problem. One way would be
to use math theory and combinatorics to enumerate all the hand rank frequencies.
Another would be to write some poker code to loop over all 2,598,960 possible
five card poker hands, and determine each hand's value if Deuces were wild.
I decided to do both, since I should get the same results using either method,
and it would act as a sanity check.

I had already written some 'C' code that determined the value of a specific
five card poker hand. It required just a little massaging to get it to
work with Deuces being wild. After lots of debugging and code checking,
I finally obtained the following results:

Poker Hand Frequencieswith Deuces Wild

672

Five of a Kind

2,552

Straight Flush

31,552

Four of a Kind

12,672

Full House

14,472

Flush

62,232

Straight

355,080

Three of a Kind

95,040

Two Pair

1,225,008

One Pair

799,680

High Card

2,598,960

TOTAL

Now the question was whether or not these figures were correct. Although
I was fairly certain my code was correct, I'd been burned before by making
invalid assumptions in my coding logic. This meant to be absolutely sure,
I needed to validate my results by using math and combinatorics.

Before we start, a quick note about combinatorics. There are often
multiple ways to obtain the same answer. In the tables given below,
I will give the combinatoric method I used to obtain my answer. You,
however, may come up with a completely different way to compute the same
answer. That is okay. It all depends on how you "count" your objects.
For example, let's say we remove all four Deuces from the deck, and we
wish to figure out how many ways we can select five cards of differing
ranks. One person might look at the problem this way:

There are 48 cards to choose from for our first card. Let's say we
select the King of Spades. For our next card, we only have 44 to choose
from (because we can't select another King). We select the Six of Diamonds.
For the third card, we only have 40 to choose from (because we can't
select any Kings or Sixes), and so on. So selecting all five cards
gives us 48 x 44 x 40 x 36 x 32 possible permutations.
However, we don't care about the order -- we want combinations instead.
In order to remove all duplicates, we must divide by 5!
(that's 5 factorial, i.e. 5x4x3x2x1). This gives a final
answer of 811,008.

Now, a second person might come along and reason as thus:

Okay, there are twelve ranks to choose from since the Deuces have
been removed. I need to choose five ranks out of the possible twelve,
so that's C(12,5), or 792 combinations. Now that I have chosen my
five distinct ranks, each of those five cards can be one of four
possible suits. So I need to multiply 792 by 45. This gives
a final answer of 811,008.

See? Two different ways to tackle the problem, but they both come up
with the same result.

Okay, enough talk. Let's start enumerating.
First off, I determined how many five card poker hands contained zero Deuces,
one Deuce, two Deuces, three Deuces, or all four Deuces. Combinatorics made
this very easy, and the following table shows the results:

Breakdown of Deuce Counts

48

hands with four Deuces

C(4,4) x C(48,1)

4,512

hands with three Deuces

C(4,3) x C(48,2)

103,776

hands with two Deuces

C(4,2) x C(48,3)

778,320

hands with one Deuce

C(4,1) x C(48,4)

1,712,304

hands with zero Deuces

C(4,0) x C(48,5)

2,598,960

TOTAL

Next, I tackled each of the five possibilities shown above. The first case
was easy. If you have all four Deuces in your five card poker hand, then
you obviously have a Five-of-a-Kind. So out of the 48 possible hands that
contain four Deuces, all of them are Five-of-a-Kinds.

Hands with Four Deuces

48

Five of a Kind

C(48,1)

48

TOTAL

Next, I took a look at the hands with three Deuces. After some thought,
you should realize that if you hold three Deuces, then the
absolute minimum hand you could hold is Four-of-a-Kind.
If the other two non-wild cards happen to be a pair, then you have a
Five-of-a-Kind. If they are the same suit and close enough in rank, you
have a Straight Flush. Otherwise, you must have a Four-of-a-Kind.
No other hands are possible with three wild Deuces.
Since there are four Deuces in a deck, but we only need to choose three of
them, each of the counts will have a factor of C(4,3) for the various
combinations of three Deuces. For Five-of-a-Kind hands, there are 12
ranks to select from (Ace through Trey) for the pair, and C(4,2) ways
to form that pair once the rank has been selected. That gives us
288 Five-of-a-Kind hands. Straight Flush hands
are more tricky to compute. We must enumerate all ways to form a
Straight Flush using two non-wild cards. It turns out there are 41 such
ways, as shown in the table below. Note that when Deuces are wild,
Six-High Straight Flushes are verboten since one can always make a higher
Straight Flush.

Straight Flushes with Three Deuces (41 ways)

Royal Flush

AK222

AQ222

AJ222

AT222

KQ222

KJ222

KT222

QJ222

QT222

JT222

King-High Straight Flush

K9222

Q9222

J9222

T9222

Queen-High Straight Flush

Q8222

J8222

T8222

98222

Jack-High Straight Flush

J7222

T7222

97222

87222

Ten-High Straight Flush

T6222

96222

86222

76222

Nine-High Straight Flush

95222

85222

75222

65222

Eight-High Straight Flush

84222

74222

64222

54222

Seven-High Straight Flush

73222

63222

53222

43222

Five-High Straight Flush

5A222

4A222

3A222

With that table, I can now calculate how many Three Deuce Straight Flush
hands there are. There are C(4,3) ways to choose three of the four Deuces,
times 41 ways to make a Straight Flush, times 4 different suits, giving
a total of 656. Once I've eliminated all Five-of-a-Kind and Straight Flush
hands, only Four-of-a-Kind are left remaining. I now know the totals
for hands with three Deuces.

Hands with Three Deuces

288

Five of a Kind

C(4,3) x 12 x C(4,2)

656

Straight Flush

C(4,3) x 41 x 4

3,568

Four of a Kind

4,512 - 288 - 656

4,512

TOTAL

Next, I worked on the hands containing two Deuces. Note that when holding
two wild Deuces, the minimum hand possible is Three-of-a-Kind. Also observe
that it is impossible to have a Full House when holding two wild cards.
The combinatorics used to calculate the hand frequencies are similar to how
we computed them for the three Deuce hands. It turns out there are 55 ways
to make a Straight Flush this time (rather than 41 as before). For Flush
hands, we select any three cards (except the Deuce) of the same suit, but
then subtract off the Straight Flush hands since those would be included.
For Straights, we use the same 55 ways to make Straight Flushes, but this
time we make sure the suits don't all match. Three-of-a-Kinds would be
a real bear to compute, so we take the easy way out. Once we've computed
all the other hand frequencies, anything left remaining has to be a
Three-of-a-Kind.

Straight Flushes with Two Deuces (55 ways)

Royal Flush

AKQ22

AKJ22

AKT22

AQJ22

AQT22

AJT22

KQJ22

KQT22

KJT22

QJT22

King-High Straight Flush

KQ922

KJ922

KT922

QJ922

QT922

JT922

Queen-High Straight Flush

QJ822

QT822

Q9822

JT822

J9822

T9822

Jack-High Straight Flush

JT722

J9722

J8722

T9722

T8722

98722

Ten-High Straight Flush

T9622

T8622

T7622

98622

97622

87622

Nine-High Straight Flush

98522

97522

96522

87522

86522

76522

Eight-High Straight Flush

87422

86422

85422

76422

75422

65422

Seven-High Straight Flush

76322

75322

74322

65322

64322

54322

Five-High Straight Flush

54A22

53A22

43A22

Hands with Two Deuces

288

Five of a Kind

C(4,2) x 12 x C(4,3)

1,320

Straight Flush

C(4,2) x 55 x 4

19,008

Four of a Kind

C(4,2) x C(4,2) x 12 x 44

0

Full House

not possible

3,960

Flush

C(4,2) x C(12,3) x 4 - 1,320

19,800

Straight

C(4,2) x 55 x 43 - 1,320

59,400

Three of a Kind

103,776 - 288 - 1,320 - 19,008 - 3,960 - 19,800

103,776

TOTAL

Almost done. Next, I compute the frequencies for hands containing
just a single wild Deuce.

Hands with One Deuce

48

Five of a Kind

4 x 12

544

Straight Flush

4 x 34 x 4

8,448

Four of a Kind

4 x C(4,3) x 12 x 44

9,504

Full House

4 x C(4,2) x C(4,2) x (12x11/2!)

7,376

Flush

4 x C(12,4) x 4 - 544

34,272

Straight

4 x 34 x 44 - 544

253,440

Three of a Kind

4 x C(4,2) x 12 x (44x40/2!)

0

Two Pair

not possible

464,688

One Pair

4 x C(12,4) x 44 - 544 - 7,376 - 34,272

0

High Card

not possible

778,320

TOTAL

Lastly, we handle all the cases where a hand has no wild cards.

Hands with No Deuces

0

Five of a Kind

not possible

32

Straight Flush

8 x 4

528

Four of a Kind

12 x 44

3,168

Full House

C(4,3) x 12 x C(4,2) x 11

3,136

Flush

4 x C(12,5) - 32

8,160

Straight

8 x 45 - 32

42,240

Three of a Kind

12 x C(4,3) x (44x40/2!)

95,040

Two Pair

(12 x C(4,2) x 11 x C(4,2))/2! x 40

760,320

One Pair

12 x C(4,2) x (44 x 40 x 36)/3!

799,680

High Card

(48 x 44 x 40 x 36 x 32)/5! - 32 - 3,136 - 8,160

1,712,304

TOTAL

Okay, that was a lot of work, but we have used raw math to enumerate
all the various hand frequencies. The final test was to see if the
totals obtained from my poker code matched the totals obtained
emperically. In other words, if my poker code states that there are
672 Five-of-a-Kind hands, then if I add up all the various ways to
make a Five-of-a-Kind using from one to four Deuces, they should be
equal.

Breakdown of Five Card Poker Handswith Deuces Wild

Hand Rank

NoDeuces

OneDeuce

TwoDeuces

ThreeDeuces

FourDeuces

SUM

Five of a Kind

0

48

288

288

48

672

Straight Flush

32

544

1,320

656

0

2,552

Four of a Kind

528

8,448

19,008

3,568

0

31,552

Full House

3,168

9,504

0

0

0

12,672

Flush

3,136

7,376

3,960

0

0

14,472

Straight

8,160

34,272

19,800

0

0

62,232

Three of a Kind

42,240

253,440

59,400

0

0

355,080

Two Pair

95,040

0

0

0

0

95,040

One Pair

760,320

464,688

0

0

0

1,225,008

High Card

799,680

0

0

0

0

799,680

TOTAL

1,712,304

778,320

103,776

4,512

48

2,598,960

At this point, it might be interesting to see how the "normal"
distribution of poker hands changes when we introduce four wild
Deuces. As expected, most of the hand count frequencies increase,
with the exception of the Two Pair and High Card hands.

Five Card Poker Hands

Nothing Wild

Deuces Wild

Hand Rank

0

672

Five of a Kind

40

2,552

Straight Flush

624

31,552

Four of a Kind

3,744

12,672

Full House

5,108

14,472

Flush

10,200

62,232

Straight

54,912

355,080

Three of a Kind

123,552

95,040

Two Pair

1,098,240

1,225,008

One Pair

1,302,540

799,680

High Card

2,598,960

2,598,960

TOTAL

One might be tempted to re-order the poker hands based on the
new frequency counts. If you did so, the Four-of-a-Kind hand
would be placed between the Flush and Straight hands, the
Three-of-a-Kind and Two Pair hands would swap places, and so
would the One Pair and High Card hands. However, in every
home game that I've played in that allows wild cards, I have
never seen the hand rankings rearranged based on the "true"
frequencies. In other words, a Flush still loses to a
Four-of-a-Kind, even thought it is actually harder
to get a Flush than a Four-of-a-Kind when Deuces are wild.
There are a number of reasons for keeping the poker hand
rankings order unchanged. First, every poker player has
already committed the "correct" ordering of poker hands to memory. Everyone
just knows that a Four-of-a-Kind is a powerful hand. Nobody
wants to be forced to remember another ordering just for
wild card games. Plus, some player would inevitably forget
the new ordering, bet his Four-of-a-Kind heavily, and then
get extremely angry when he loses to a Flush or a Full House.
Second, one could always "cheat" by changing the value of
your hand. For example, suppose you hold three wild Deuces
along with the Six and Jack of Spades. Logic would tell
you that you have Four Jacks with a Six kicker. However,
since Full Houses are supposedly harder to get than Four-of-a-Kinds,
you would instead say that you have a Full House, Jacks over
Sixes. It's sort of a Catch-22. If you rearrange the poker
hand orders, then players will just start calling their hands
differently to circumvent the new ordering, which in turn will
screw up the frequencies again, which sort of defeats the
purpose of re-ordering the hands in the first place!
I guess one could get around this trickery (of a hand
having multiple values) by enforcing a "rule" that a player
with wild cards must call his hand as high as possible based
on the "normal" frequencies, even if that means having a
losing hand; but that also would be hard to remember and
cause lots of frustration at showdown. Third, one of the
main reasons of having wild card games is the chance to gleefully
announce that you have a Four-of-a-Kind or a Straight Flush.
Let's face it, that doesn't happen too often in regular poker.
It's great fun to be able to lay down and win with one of
these "power" hands, which wouldn't happen as often if we put the
Four-of-a-Kind in its "proper" place. Lastly, consider the
game of Seven Card Stud (nothing wild). Each player at showdown
will have seven cards, of which they each make the best five
card hand possible. If you calculate the hand frequencies of
all possible C(52,7) combinations, and order them based on their
likelihood of appearing, you get the exact same order as we did
with just five cards, except that One Pair is easier to
get than a High Card hand. It's actually easier to get a pair
than seven crappy cards (which explains why the game Razz is
so much fun), but the poker community didn't try
to reverse the hand rankings of those two hands when playing
Seven Card Stud. They remained the same. So likewise, even though
wild cards throw the hand rankings all out of whack, the "normal"
ordering is maintained.

Well, I hope you enjoyed this geeky and math-laden journey into
the world of poker. I doubt if this information will make you
a better poker player, and unless you play in a home game, you
probably won't be playing too many games containing wild cards.